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\begin{frontmatter}

%\title{Applications and improvements of the PFEM (particles + finite elements) methodology to free surface flows.}
% \title{A free surface extension of semiLagrangian particle and finite element methodologies.}
% \title{Extension and validation of the PFEM methodology to free surface flows}
%\title{Extension and validation of a last generation of PFEM on free-surface flows}
\title{An extended validation of the last generation of particle finite element method for free surface flows}
%\title{A complete validation of the particle finite element method for free surface flows}
%\title{An efficient adaptation of the particle finite element method to free surface flows}
\tnotetext[mytitlenote]{Fully documented templates are available in the elsarticle package on
 \href{http://www.ctan.org/tex-archive/macros/latex/contrib/elsarticle}{CTAN}.}

%% Group authors per affiliation:
% \author{Juan M. Gimenez\fnref{myfootnote}}
% \address{CIMEC, Santa Fe, Argentina}
% \author{Leo M. Gonz\'{a}lez\fnref{myfootnote}}
% \address{UPM, Madrid, Spain}
%\fntext[myfootnote]{Since 1880.}

% or include affiliations in footnotes:
\author[mymainaddress]{Juan M. Gimenez\corref{mycorrespondingauthor}}
\cortext[mycorrespondingauthor]{Corresponding author}
\ead{jmarcelogimenez@gmail.com}
%\ead[url]{www.elsevier.com}
\author[mysecondaryaddress]{Leo M. Gonz\'{a}lez}
\address[mymainaddress]{Centro de Investigaci\'on de M\'etodos Computacionales (CIMEC) - UNL/CONICET, Predio Conicet-Santa Fe Colectora Ruta Nac 168
	      Paraje El Pozo, Santa Fe, Argentina.}
\address[mysecondaryaddress]{Escuela T\'{e}cnica Superior de Ingenieros Navales, Universidad Polit\'{e}cnica de Madrid (ETSIN-UPM), Avd. Arco de la Victoria 4, Madrid, Spain}

\begin{abstract}
%In this paper, a new generation of the particle method known as Particle Finite Element Method (PFEM), which combines convective particle movement and a fixed mesh resolution, is applied to free surface flows. This methodology, named PFEM-2, presents two novel steps: first, the possibility of using larger time steps, compared to other similar numerical tools, which shows that shorter computational times can be achieved while maintaining the accuracy of the solution in a wide variety of problems. Second, since surface flows are the main topic of this paper, different improved versions of discontinuous and continuous enriched basis functions for the pressure field have also been developed, thus reconstructing the free surface without artificial diffusion or undesired numerical effects. Combining these two improvements, a variety of free surface flows have been solved in 2D and 3D cases, where the evident advantages of the improvements are remarked. The collection of problems has been carefully selected such that a wide variety of Froude numbers, density ratios and dominant dissipative cases are presented with the intention of presenting a general methodology, not restricted to a particular range of applications. The results of the different free-surface problems solved, which include: Rayleigh-Taylor instability, sloshing problems, viscous standing waves and the dam break problem, are compared to well validated numerical alternatives and experimental measurements and good approximations for such complex flows were obtained.
In this paper, a new generation of the particle method known as Particle Finite Element Method (PFEM), which combines convective particle movement and a fixed mesh resolution, is applied to free surface flows. This interesting variant, previously described in the literature as PFEM-2, is able to use larger time steps when compared to other similar numerical tools which implies shorter computational times while maintaining the accuracy of the computation. PFEM-2 has already been extended to free surface problems, being the main topic of this paper a deep validation of this methodology for a wider range of flows. To accomplish this task, different improved versions of discontinuous and continuous enriched basis functions for the pressure field have been developed to capture the free surface dynamics without artificial diffusion or undesired numerical effects when different density ratios are involved. A collection of problems has been carefully selected such that a wide variety of Froude numbers, density ratios and dominant dissipative cases are reported with the intention of presenting a general methodology, not restricted to a particular range of parameters, and capable of using large time-steps. The results of the different free-surface problems solved, which include: Rayleigh-Taylor instability, sloshing problems, viscous standing waves and the dam break problem, are compared to well validated numerical alternatives or experimental measurements obtaining accurate approximations for such complex flows.
\end{abstract}

\begin{keyword}
PFEM; PFEM-2; free surface flows; finite elements; large time-steps; enrichment
% \MSC[2010] 00-01\sep  99-00
\end{keyword}

\end{frontmatter}

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\fboxsep=0mm%padding thickness

\section{Introduction}\label{Intro}
\input{Intro}


\section{PFEM-2 Algorithm}\label{PFEM_Algorithm}
% \input{PFEM}
\input{PFEM_r1}

\section[Free-Surface treatment]{Free-Surface flows treatment}\label{Free_surface}
% \input{Free_surface}
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\input{PFEM_multifluids}

% \section{Second order improvements.}\label{Second_order}
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\section{Results}\label{FS_results}

In this section, a wide range of free-surface problems with different ratio between densities and viscosities of the fluids involved are solved using the PFEM-2 method and the results compared with reference ones, which include numerical results, experimental analysis and/or analytical solutions.

The first case is the widely known Rayleigh-Taylor instability, where a small perturbation must generate complex fluid structures that are well reported in literature.
This problem focuses on the capabilities of the method to deal with large time-steps. These results are compared with those obtained with a well reputed Eulerian strategy, named Volume of Fluid (VoF), which adds limiters as a method of guaranteeing boundedness of phase-fractions, and interface compression numerical terms to keep the interface sharp.
The second test is a sloshing problem which allows the numerical strategy behavior to be tested when different density ratios are simulated, the upcoming results are compared to well validated codes. In this second test, a discussion about the enrichment strategies is presented. Next, the ability of the method to deal with a highly dissipative free surface flow is tested with a standing wave problem, results are compared to semi-analytical solutions of the decayment of the total energy of the system.
Finally, the last example is a direct comparison against results coming from a dam-break experiment.
This case shows that the method can solve large motions of the interfaces and splashing of waves, while maintaining acceptable levels of accuracy when a comparison against pressure and height measurements is performed.

The main aim of this section is to show the capability of the method to work with large Courant numbers without stability loss and with negligible resign accuracy, reasonable large time-steps are then selected for each test. Previous PFEM-2 works (see \cite{Idelsohn12b}\cite{Gimenez14}), showed that the computational cost of each time-step is almost equal to traditional Eulerian solvers, and the increase of time-step decreases the duration of the global computation without loss of accuracy. Herein, the efficient distributed-memory implementation presented in \cite{Gimenez14} is extended to the free-surface treatment and used to simulate each of next cases presented.

 %if the method is able to solve simulation using larger time-steps than another codes, consequently it will be faster to solve the same problem and it will obtain results with guaranteed accuracy.

\input{FS_results_RT}
\input{FS_results_Ansari}
\input{FS_results_standingwave}
\input{FS_results_DamBreak}
% \input{FS_results_Roll}


\section{Conclusions}

The last generation of the Particle Finite Element Method (PFEM-2) is a contemporary strategy which uses a spatial discretization based on a background mesh and a cloud of particles. The dynamics equations are solved in a Lagrangian frame, where the implicit nonlinearities of the equation are solved using the {X-IVAS} strategy. That explicit temporal integration for convective terms allows for the use of large time-steps, thus providing a very efficient way when computing times are concerned.

% In the current work, a general formulation to solve free-surface flows with pressure gradient discontinuities was presented and exhaustively tested. That algorithm, which is based on a continuous enriched space for pressure, has shown good accuracy when solving a wide range of multiphase problems and keeping the advantage of the possibility to use large time-steps. Several cases with a large variety of Froude numbers, density ratios and dominant dissipative cases have also been analyzed. These results were compared with other reference softwares, semi-analytical expressions and also experimental data. In each of them, PFEM-2 has proven to be accurately competitive and computationally efficient. Regarding CPU times, they can be decreased without accuracy loss if the condensing strategy for enriched pressure degrees of freedom is used. Although that approach loses the generality of the formulation for any range of application, the computational cost enforcing the asseveration is reduced, which, to our knowledge, currently makes PFEM-2 the faster algorithm for solving multiphase flows.

In the current work, a formulation to solve free-surface flows with pressure gradient discontinuities, presented in \cite{Idelsohn13c}, was generalized and exhaustively tested. That algorithm, which is based on a continuous enriched space for pressure, has shown good accuracy when solving a wide range of multiphase problems and keeping the advantage of the possibility to use large time-steps. Several cases with a large variety of Froude numbers, density ratios and dominant dissipative cases have also been analyzed. These results were compared with other reference softwares, semi-analytical expressions and also experimental data. In each of them, PFEM-2 has proven to be accurately competitive and computationally efficient. Regarding CPU times, they can be decreased without accuracy loss if the original condensing strategy for enriched pressure degrees of freedom is used. Although that approach loses the generality of the formulation for any range of application, the computational cost is reduced, which, to our knowledge, currently makes PFEM-2 the faster algorithm for solving two phase flows.
% use section* for acknowledgement
\section*{Acknowledgment}

Authors thank to Pedro Gal\'an del Sastre (UPM-Spain), Eng. Pablo Becker (CIMNE-Spain), Dr. Norberto Nigro (CIMEC-Argentina) and Prof. Sergio Idelsohn (CIMNE-Spain and CIMEC-Argentina) for their valuable help and discussions about the method features and its implementation. Also to Dr. Santiago M\'arquez Dami\'an (CIMEC-Argentina) for his contribution in OpenFOAM simulations.

J. Gimenez gratefully acknowledges the support of the Argentinian Agencia Nacional de Promoci\'on Cient\'ifica y T\' ecnica (ANPCyT) through a doctoral grant in the FONCyT program. Also the authors would like to thank the program ERASMUS mundus action 2 ARCOIRIS project for their financial support throught a six-month doctoral scholarship.

Other financial support was provided by CONICET, Universidad Nacional del Litoral (CAI+D Tipo II 65-333 (2009)), ANPCyT-FONCyT (grants PICT 1645 BID (2008)) and ERC Advanced Grant REALTIME project AdG-2009325.

The research leading to these results has also received funding from the Spanish Ministry for Science and Innovation under
grant TRA2010-16988 ``\textit{Op\-ti\-mi\-za\-ci\'on del transporte de Gas Licuado en buques LNG mediante estudios sobre in\-te\-rac\-ci\'on fluido-estructura}'' .

All the authors want to thank Mr. Hugo Gee for his valuable assistance during the preparation of this manuscript.

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